Ideal flat bands in Twisted Bilayer Graphene with substrate

We consider twisted bilayer graphene with one layer stacked on top of a commensurate substrate with triangular Bravais lattice, which induces an inter-valley coupling and spin-orbital coupling. The substrate introduces a variety of couplings that support topologically non-trivial flat bands near the Fermi level, with spin Chern numbers $|\mathcal{C}|=1, 2$. These topological flat bands remain nontrivial in substrates with $C_{2z}$-breaking sublattices. We further show that the quantum geometry of these flat bands are close to the so-called ideal bands, which gives platforms for potentially realizing fractional Chern insulator phases when interactions are introduced.